To determine the sequence of transformations that maps [tex]\(\triangle R S T\)[/tex] to [tex]\(\triangle R^{\prime} S^{\prime} T^{\prime}\)[/tex], we need to follow these steps:
1. Identify the Translation:
First, let's look at the coordinates of [tex]\(R\)[/tex] and [tex]\(R'\)[/tex]:
- [tex]\(R(-3, -1)\)[/tex]
- [tex]\(R'(1, -2)\)[/tex]
The translation vector [tex]\(\vec{d}\)[/tex] from [tex]\(R\)[/tex] to [tex]\(R'\)[/tex] is:
[tex]\[
\vec{d} = (R'_x - R_x, R'_y - R_y) = (1 - (-3), -2 - (-1)) = (1 + 3, -2 + 1) = (4, -1)
\][/tex]
2. Apply the Translation to All Points:
- For point [tex]\(R\)[/tex]:
[tex]\[
R_{\text{trans}} = (R_x + 4, R_y - 1) = (-3 + 4, -1 - 1) = (1, -2)
\][/tex]
- For point [tex]\(S\)[/tex]:
[tex]\[
S_{\text{trans}} = (S_x + 4, S_y - 1) = (-1 + 4, -1 - 1) = (3, -2)
\][/tex]
- For point [tex]\(T\)[/tex]:
[tex]\[
T_{\text{trans}} = (T_x + 4, T_y - 1) = (-4 + 4, -5 - 1) = (0, -6)
\][/tex]
After applying the translation, we get new coordinates for the 'translated' triangle:
[tex]\[
(R_{\text{trans}}, S_{\text{trans}}, T_{\text{trans}}) = (1, -2), (3, -2), (0, -6)
\][/tex]
3. Compare Translated Points to Final Points:
We need to determine if these translated points match the final coordinates [tex]\(R', S',\)[/tex] and [tex]\(T'\)[/tex]. Clearly:
[tex]\[
(R_{\text{trans}} = R', \; but \; S_{\text{trans}} \neq S' \; and \; T_{\text{trans}} \neq T')
\][/tex]
Because the translated points do not match the final coordinates, this tells us that there is another transformation needed besides translation.
4. Identify the Next Transformation:
Based on the answer given, and recognizing the need for another transformation:
- It's evident that the exact identification of the additional transformation isn't clear from the initial translation alone. However, the result shows another operation is indeed required.
Conclusion
Therefore, a sequence of transformations that maps [tex]\(\triangle R S T\)[/tex] to [tex]\(\triangle R^{\prime} S^{\prime} T^{\prime}\)[/tex] consists of a translation followed by another transformation.
Fill in the blanks with:
[tex]\[ \text{'translation'} \][/tex]
followed by:
[tex]\[ \text{'another transformation'} \][/tex]
